Properties

Label 4864.2.a.bs.1.8
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 23 x^{8} + 44 x^{7} + 167 x^{6} - 266 x^{5} - 491 x^{4} + 460 x^{3} + 546 x^{2} + 56 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.42960\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.15181 q^{3} +1.07587 q^{5} +0.238565 q^{7} -1.67333 q^{9} +O(q^{10})\) \(q+1.15181 q^{3} +1.07587 q^{5} +0.238565 q^{7} -1.67333 q^{9} +2.84249 q^{11} -1.31444 q^{13} +1.23921 q^{15} -4.47279 q^{17} -1.00000 q^{19} +0.274782 q^{21} +1.94338 q^{23} -3.84249 q^{25} -5.38280 q^{27} -8.83944 q^{29} -6.64877 q^{31} +3.27402 q^{33} +0.256666 q^{35} +4.06763 q^{37} -1.51399 q^{39} -2.78211 q^{41} +1.21363 q^{43} -1.80029 q^{45} -10.2695 q^{47} -6.94309 q^{49} -5.15181 q^{51} +8.36231 q^{53} +3.05817 q^{55} -1.15181 q^{57} -9.25916 q^{59} +4.66035 q^{61} -0.399197 q^{63} -1.41417 q^{65} -4.31849 q^{67} +2.23841 q^{69} +2.78069 q^{71} -0.134460 q^{73} -4.42583 q^{75} +0.678119 q^{77} +12.7841 q^{79} -1.17998 q^{81} -13.4028 q^{83} -4.81216 q^{85} -10.1814 q^{87} +9.20650 q^{89} -0.313579 q^{91} -7.65814 q^{93} -1.07587 q^{95} -0.247499 q^{97} -4.75643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{3} + 14q^{9} + O(q^{10}) \) \( 10q - 4q^{3} + 14q^{9} - 20q^{11} + 4q^{17} - 10q^{19} + 10q^{25} - 28q^{27} - 8q^{33} - 36q^{35} - 12q^{41} + 4q^{43} + 26q^{49} - 36q^{51} + 4q^{57} - 52q^{59} - 24q^{65} - 12q^{67} + 12q^{73} + 12q^{75} + 34q^{81} - 16q^{83} - 20q^{89} - 60q^{91} - 28q^{97} - 60q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.15181 0.664999 0.332500 0.943103i \(-0.392108\pi\)
0.332500 + 0.943103i \(0.392108\pi\)
\(4\) 0 0
\(5\) 1.07587 0.481146 0.240573 0.970631i \(-0.422665\pi\)
0.240573 + 0.970631i \(0.422665\pi\)
\(6\) 0 0
\(7\) 0.238565 0.0901690 0.0450845 0.998983i \(-0.485644\pi\)
0.0450845 + 0.998983i \(0.485644\pi\)
\(8\) 0 0
\(9\) −1.67333 −0.557776
\(10\) 0 0
\(11\) 2.84249 0.857044 0.428522 0.903531i \(-0.359034\pi\)
0.428522 + 0.903531i \(0.359034\pi\)
\(12\) 0 0
\(13\) −1.31444 −0.364560 −0.182280 0.983247i \(-0.558348\pi\)
−0.182280 + 0.983247i \(0.558348\pi\)
\(14\) 0 0
\(15\) 1.23921 0.319962
\(16\) 0 0
\(17\) −4.47279 −1.08481 −0.542405 0.840117i \(-0.682486\pi\)
−0.542405 + 0.840117i \(0.682486\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.274782 0.0599623
\(22\) 0 0
\(23\) 1.94338 0.405222 0.202611 0.979259i \(-0.435057\pi\)
0.202611 + 0.979259i \(0.435057\pi\)
\(24\) 0 0
\(25\) −3.84249 −0.768499
\(26\) 0 0
\(27\) −5.38280 −1.03592
\(28\) 0 0
\(29\) −8.83944 −1.64144 −0.820721 0.571329i \(-0.806428\pi\)
−0.820721 + 0.571329i \(0.806428\pi\)
\(30\) 0 0
\(31\) −6.64877 −1.19415 −0.597077 0.802184i \(-0.703671\pi\)
−0.597077 + 0.802184i \(0.703671\pi\)
\(32\) 0 0
\(33\) 3.27402 0.569933
\(34\) 0 0
\(35\) 0.256666 0.0433845
\(36\) 0 0
\(37\) 4.06763 0.668715 0.334357 0.942446i \(-0.391481\pi\)
0.334357 + 0.942446i \(0.391481\pi\)
\(38\) 0 0
\(39\) −1.51399 −0.242432
\(40\) 0 0
\(41\) −2.78211 −0.434492 −0.217246 0.976117i \(-0.569707\pi\)
−0.217246 + 0.976117i \(0.569707\pi\)
\(42\) 0 0
\(43\) 1.21363 0.185077 0.0925386 0.995709i \(-0.470502\pi\)
0.0925386 + 0.995709i \(0.470502\pi\)
\(44\) 0 0
\(45\) −1.80029 −0.268372
\(46\) 0 0
\(47\) −10.2695 −1.49796 −0.748978 0.662595i \(-0.769455\pi\)
−0.748978 + 0.662595i \(0.769455\pi\)
\(48\) 0 0
\(49\) −6.94309 −0.991870
\(50\) 0 0
\(51\) −5.15181 −0.721398
\(52\) 0 0
\(53\) 8.36231 1.14865 0.574326 0.818627i \(-0.305264\pi\)
0.574326 + 0.818627i \(0.305264\pi\)
\(54\) 0 0
\(55\) 3.05817 0.412363
\(56\) 0 0
\(57\) −1.15181 −0.152561
\(58\) 0 0
\(59\) −9.25916 −1.20544 −0.602720 0.797953i \(-0.705916\pi\)
−0.602720 + 0.797953i \(0.705916\pi\)
\(60\) 0 0
\(61\) 4.66035 0.596697 0.298349 0.954457i \(-0.403564\pi\)
0.298349 + 0.954457i \(0.403564\pi\)
\(62\) 0 0
\(63\) −0.399197 −0.0502941
\(64\) 0 0
\(65\) −1.41417 −0.175407
\(66\) 0 0
\(67\) −4.31849 −0.527587 −0.263794 0.964579i \(-0.584974\pi\)
−0.263794 + 0.964579i \(0.584974\pi\)
\(68\) 0 0
\(69\) 2.23841 0.269472
\(70\) 0 0
\(71\) 2.78069 0.330007 0.165003 0.986293i \(-0.447236\pi\)
0.165003 + 0.986293i \(0.447236\pi\)
\(72\) 0 0
\(73\) −0.134460 −0.0157373 −0.00786866 0.999969i \(-0.502505\pi\)
−0.00786866 + 0.999969i \(0.502505\pi\)
\(74\) 0 0
\(75\) −4.42583 −0.511051
\(76\) 0 0
\(77\) 0.678119 0.0772788
\(78\) 0 0
\(79\) 12.7841 1.43832 0.719162 0.694843i \(-0.244526\pi\)
0.719162 + 0.694843i \(0.244526\pi\)
\(80\) 0 0
\(81\) −1.17998 −0.131109
\(82\) 0 0
\(83\) −13.4028 −1.47115 −0.735573 0.677445i \(-0.763087\pi\)
−0.735573 + 0.677445i \(0.763087\pi\)
\(84\) 0 0
\(85\) −4.81216 −0.521952
\(86\) 0 0
\(87\) −10.1814 −1.09156
\(88\) 0 0
\(89\) 9.20650 0.975887 0.487944 0.872875i \(-0.337747\pi\)
0.487944 + 0.872875i \(0.337747\pi\)
\(90\) 0 0
\(91\) −0.313579 −0.0328720
\(92\) 0 0
\(93\) −7.65814 −0.794112
\(94\) 0 0
\(95\) −1.07587 −0.110382
\(96\) 0 0
\(97\) −0.247499 −0.0251297 −0.0125648 0.999921i \(-0.504000\pi\)
−0.0125648 + 0.999921i \(0.504000\pi\)
\(98\) 0 0
\(99\) −4.75643 −0.478039
\(100\) 0 0
\(101\) 16.8026 1.67193 0.835963 0.548786i \(-0.184910\pi\)
0.835963 + 0.548786i \(0.184910\pi\)
\(102\) 0 0
\(103\) 5.40957 0.533020 0.266510 0.963832i \(-0.414129\pi\)
0.266510 + 0.963832i \(0.414129\pi\)
\(104\) 0 0
\(105\) 0.295631 0.0288506
\(106\) 0 0
\(107\) −6.39113 −0.617853 −0.308927 0.951086i \(-0.599970\pi\)
−0.308927 + 0.951086i \(0.599970\pi\)
\(108\) 0 0
\(109\) −14.8779 −1.42505 −0.712524 0.701648i \(-0.752448\pi\)
−0.712524 + 0.701648i \(0.752448\pi\)
\(110\) 0 0
\(111\) 4.68515 0.444695
\(112\) 0 0
\(113\) 8.59073 0.808148 0.404074 0.914726i \(-0.367594\pi\)
0.404074 + 0.914726i \(0.367594\pi\)
\(114\) 0 0
\(115\) 2.09083 0.194971
\(116\) 0 0
\(117\) 2.19949 0.203343
\(118\) 0 0
\(119\) −1.06705 −0.0978163
\(120\) 0 0
\(121\) −2.92023 −0.265476
\(122\) 0 0
\(123\) −3.20446 −0.288937
\(124\) 0 0
\(125\) −9.51342 −0.850906
\(126\) 0 0
\(127\) −1.15983 −0.102918 −0.0514592 0.998675i \(-0.516387\pi\)
−0.0514592 + 0.998675i \(0.516387\pi\)
\(128\) 0 0
\(129\) 1.39788 0.123076
\(130\) 0 0
\(131\) 3.85559 0.336864 0.168432 0.985713i \(-0.446130\pi\)
0.168432 + 0.985713i \(0.446130\pi\)
\(132\) 0 0
\(133\) −0.238565 −0.0206862
\(134\) 0 0
\(135\) −5.79122 −0.498428
\(136\) 0 0
\(137\) −1.42976 −0.122152 −0.0610761 0.998133i \(-0.519453\pi\)
−0.0610761 + 0.998133i \(0.519453\pi\)
\(138\) 0 0
\(139\) −3.58190 −0.303813 −0.151907 0.988395i \(-0.548541\pi\)
−0.151907 + 0.988395i \(0.548541\pi\)
\(140\) 0 0
\(141\) −11.8285 −0.996139
\(142\) 0 0
\(143\) −3.73629 −0.312444
\(144\) 0 0
\(145\) −9.51013 −0.789773
\(146\) 0 0
\(147\) −7.99713 −0.659592
\(148\) 0 0
\(149\) 16.2763 1.33341 0.666705 0.745322i \(-0.267704\pi\)
0.666705 + 0.745322i \(0.267704\pi\)
\(150\) 0 0
\(151\) −14.3242 −1.16569 −0.582845 0.812583i \(-0.698061\pi\)
−0.582845 + 0.812583i \(0.698061\pi\)
\(152\) 0 0
\(153\) 7.48445 0.605082
\(154\) 0 0
\(155\) −7.15325 −0.574563
\(156\) 0 0
\(157\) 3.73495 0.298081 0.149041 0.988831i \(-0.452381\pi\)
0.149041 + 0.988831i \(0.452381\pi\)
\(158\) 0 0
\(159\) 9.63181 0.763852
\(160\) 0 0
\(161\) 0.463621 0.0365385
\(162\) 0 0
\(163\) 3.98861 0.312412 0.156206 0.987724i \(-0.450074\pi\)
0.156206 + 0.987724i \(0.450074\pi\)
\(164\) 0 0
\(165\) 3.52243 0.274221
\(166\) 0 0
\(167\) 8.17158 0.632336 0.316168 0.948703i \(-0.397604\pi\)
0.316168 + 0.948703i \(0.397604\pi\)
\(168\) 0 0
\(169\) −11.2722 −0.867096
\(170\) 0 0
\(171\) 1.67333 0.127963
\(172\) 0 0
\(173\) 0.351356 0.0267131 0.0133565 0.999911i \(-0.495748\pi\)
0.0133565 + 0.999911i \(0.495748\pi\)
\(174\) 0 0
\(175\) −0.916684 −0.0692948
\(176\) 0 0
\(177\) −10.6648 −0.801616
\(178\) 0 0
\(179\) −15.2740 −1.14163 −0.570817 0.821077i \(-0.693373\pi\)
−0.570817 + 0.821077i \(0.693373\pi\)
\(180\) 0 0
\(181\) 8.94630 0.664974 0.332487 0.943108i \(-0.392112\pi\)
0.332487 + 0.943108i \(0.392112\pi\)
\(182\) 0 0
\(183\) 5.36785 0.396803
\(184\) 0 0
\(185\) 4.37626 0.321749
\(186\) 0 0
\(187\) −12.7139 −0.929730
\(188\) 0 0
\(189\) −1.28415 −0.0934079
\(190\) 0 0
\(191\) 18.3946 1.33098 0.665492 0.746405i \(-0.268222\pi\)
0.665492 + 0.746405i \(0.268222\pi\)
\(192\) 0 0
\(193\) −5.09902 −0.367035 −0.183518 0.983016i \(-0.558748\pi\)
−0.183518 + 0.983016i \(0.558748\pi\)
\(194\) 0 0
\(195\) −1.62886 −0.116645
\(196\) 0 0
\(197\) −20.2916 −1.44572 −0.722860 0.690995i \(-0.757173\pi\)
−0.722860 + 0.690995i \(0.757173\pi\)
\(198\) 0 0
\(199\) 0.310999 0.0220461 0.0110231 0.999939i \(-0.496491\pi\)
0.0110231 + 0.999939i \(0.496491\pi\)
\(200\) 0 0
\(201\) −4.97408 −0.350845
\(202\) 0 0
\(203\) −2.10878 −0.148007
\(204\) 0 0
\(205\) −2.99320 −0.209054
\(206\) 0 0
\(207\) −3.25191 −0.226023
\(208\) 0 0
\(209\) −2.84249 −0.196619
\(210\) 0 0
\(211\) −14.3284 −0.986406 −0.493203 0.869914i \(-0.664174\pi\)
−0.493203 + 0.869914i \(0.664174\pi\)
\(212\) 0 0
\(213\) 3.20283 0.219454
\(214\) 0 0
\(215\) 1.30572 0.0890491
\(216\) 0 0
\(217\) −1.58616 −0.107676
\(218\) 0 0
\(219\) −0.154872 −0.0104653
\(220\) 0 0
\(221\) 5.87921 0.395479
\(222\) 0 0
\(223\) −12.8621 −0.861312 −0.430656 0.902516i \(-0.641718\pi\)
−0.430656 + 0.902516i \(0.641718\pi\)
\(224\) 0 0
\(225\) 6.42976 0.428650
\(226\) 0 0
\(227\) −18.4491 −1.22451 −0.612256 0.790659i \(-0.709738\pi\)
−0.612256 + 0.790659i \(0.709738\pi\)
\(228\) 0 0
\(229\) −4.73279 −0.312751 −0.156376 0.987698i \(-0.549981\pi\)
−0.156376 + 0.987698i \(0.549981\pi\)
\(230\) 0 0
\(231\) 0.781066 0.0513903
\(232\) 0 0
\(233\) −4.27809 −0.280267 −0.140133 0.990133i \(-0.544753\pi\)
−0.140133 + 0.990133i \(0.544753\pi\)
\(234\) 0 0
\(235\) −11.0487 −0.720735
\(236\) 0 0
\(237\) 14.7249 0.956483
\(238\) 0 0
\(239\) 20.3905 1.31895 0.659474 0.751727i \(-0.270779\pi\)
0.659474 + 0.751727i \(0.270779\pi\)
\(240\) 0 0
\(241\) 25.5877 1.64825 0.824123 0.566410i \(-0.191668\pi\)
0.824123 + 0.566410i \(0.191668\pi\)
\(242\) 0 0
\(243\) 14.7893 0.948732
\(244\) 0 0
\(245\) −7.46989 −0.477234
\(246\) 0 0
\(247\) 1.31444 0.0836358
\(248\) 0 0
\(249\) −15.4375 −0.978311
\(250\) 0 0
\(251\) −28.9847 −1.82950 −0.914749 0.404024i \(-0.867611\pi\)
−0.914749 + 0.404024i \(0.867611\pi\)
\(252\) 0 0
\(253\) 5.52404 0.347293
\(254\) 0 0
\(255\) −5.54270 −0.347098
\(256\) 0 0
\(257\) 5.10203 0.318256 0.159128 0.987258i \(-0.449132\pi\)
0.159128 + 0.987258i \(0.449132\pi\)
\(258\) 0 0
\(259\) 0.970394 0.0602974
\(260\) 0 0
\(261\) 14.7913 0.915558
\(262\) 0 0
\(263\) 11.6953 0.721160 0.360580 0.932728i \(-0.382579\pi\)
0.360580 + 0.932728i \(0.382579\pi\)
\(264\) 0 0
\(265\) 8.99680 0.552669
\(266\) 0 0
\(267\) 10.6042 0.648964
\(268\) 0 0
\(269\) 0.163433 0.00996466 0.00498233 0.999988i \(-0.498414\pi\)
0.00498233 + 0.999988i \(0.498414\pi\)
\(270\) 0 0
\(271\) 1.01216 0.0614846 0.0307423 0.999527i \(-0.490213\pi\)
0.0307423 + 0.999527i \(0.490213\pi\)
\(272\) 0 0
\(273\) −0.361184 −0.0218599
\(274\) 0 0
\(275\) −10.9223 −0.658637
\(276\) 0 0
\(277\) −3.14825 −0.189160 −0.0945800 0.995517i \(-0.530151\pi\)
−0.0945800 + 0.995517i \(0.530151\pi\)
\(278\) 0 0
\(279\) 11.1256 0.666071
\(280\) 0 0
\(281\) 24.6734 1.47189 0.735946 0.677040i \(-0.236738\pi\)
0.735946 + 0.677040i \(0.236738\pi\)
\(282\) 0 0
\(283\) 13.8095 0.820889 0.410445 0.911886i \(-0.365374\pi\)
0.410445 + 0.911886i \(0.365374\pi\)
\(284\) 0 0
\(285\) −1.23921 −0.0734042
\(286\) 0 0
\(287\) −0.663713 −0.0391777
\(288\) 0 0
\(289\) 3.00584 0.176814
\(290\) 0 0
\(291\) −0.285072 −0.0167112
\(292\) 0 0
\(293\) −8.24682 −0.481784 −0.240892 0.970552i \(-0.577440\pi\)
−0.240892 + 0.970552i \(0.577440\pi\)
\(294\) 0 0
\(295\) −9.96169 −0.579992
\(296\) 0 0
\(297\) −15.3006 −0.887829
\(298\) 0 0
\(299\) −2.55445 −0.147728
\(300\) 0 0
\(301\) 0.289530 0.0166882
\(302\) 0 0
\(303\) 19.3535 1.11183
\(304\) 0 0
\(305\) 5.01396 0.287098
\(306\) 0 0
\(307\) −17.4540 −0.996153 −0.498076 0.867133i \(-0.665960\pi\)
−0.498076 + 0.867133i \(0.665960\pi\)
\(308\) 0 0
\(309\) 6.23080 0.354458
\(310\) 0 0
\(311\) 10.5256 0.596851 0.298425 0.954433i \(-0.403539\pi\)
0.298425 + 0.954433i \(0.403539\pi\)
\(312\) 0 0
\(313\) 14.3498 0.811098 0.405549 0.914073i \(-0.367080\pi\)
0.405549 + 0.914073i \(0.367080\pi\)
\(314\) 0 0
\(315\) −0.429486 −0.0241988
\(316\) 0 0
\(317\) −16.3470 −0.918138 −0.459069 0.888401i \(-0.651817\pi\)
−0.459069 + 0.888401i \(0.651817\pi\)
\(318\) 0 0
\(319\) −25.1260 −1.40679
\(320\) 0 0
\(321\) −7.36138 −0.410872
\(322\) 0 0
\(323\) 4.47279 0.248873
\(324\) 0 0
\(325\) 5.05073 0.280164
\(326\) 0 0
\(327\) −17.1366 −0.947656
\(328\) 0 0
\(329\) −2.44993 −0.135069
\(330\) 0 0
\(331\) 15.0118 0.825123 0.412562 0.910930i \(-0.364634\pi\)
0.412562 + 0.910930i \(0.364634\pi\)
\(332\) 0 0
\(333\) −6.80649 −0.372993
\(334\) 0 0
\(335\) −4.64615 −0.253846
\(336\) 0 0
\(337\) 0.462187 0.0251769 0.0125885 0.999921i \(-0.495993\pi\)
0.0125885 + 0.999921i \(0.495993\pi\)
\(338\) 0 0
\(339\) 9.89491 0.537418
\(340\) 0 0
\(341\) −18.8991 −1.02344
\(342\) 0 0
\(343\) −3.32633 −0.179605
\(344\) 0 0
\(345\) 2.40824 0.129655
\(346\) 0 0
\(347\) −17.2669 −0.926935 −0.463468 0.886114i \(-0.653395\pi\)
−0.463468 + 0.886114i \(0.653395\pi\)
\(348\) 0 0
\(349\) 26.1466 1.39960 0.699798 0.714341i \(-0.253273\pi\)
0.699798 + 0.714341i \(0.253273\pi\)
\(350\) 0 0
\(351\) 7.07536 0.377655
\(352\) 0 0
\(353\) 12.3930 0.659614 0.329807 0.944048i \(-0.393016\pi\)
0.329807 + 0.944048i \(0.393016\pi\)
\(354\) 0 0
\(355\) 2.99167 0.158781
\(356\) 0 0
\(357\) −1.22904 −0.0650478
\(358\) 0 0
\(359\) −14.1664 −0.747673 −0.373837 0.927495i \(-0.621958\pi\)
−0.373837 + 0.927495i \(0.621958\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.36356 −0.176541
\(364\) 0 0
\(365\) −0.144662 −0.00757195
\(366\) 0 0
\(367\) 3.00328 0.156770 0.0783850 0.996923i \(-0.475024\pi\)
0.0783850 + 0.996923i \(0.475024\pi\)
\(368\) 0 0
\(369\) 4.65538 0.242349
\(370\) 0 0
\(371\) 1.99495 0.103573
\(372\) 0 0
\(373\) −38.0432 −1.96980 −0.984901 0.173119i \(-0.944615\pi\)
−0.984901 + 0.173119i \(0.944615\pi\)
\(374\) 0 0
\(375\) −10.9577 −0.565852
\(376\) 0 0
\(377\) 11.6189 0.598404
\(378\) 0 0
\(379\) −28.6291 −1.47058 −0.735290 0.677753i \(-0.762954\pi\)
−0.735290 + 0.677753i \(0.762954\pi\)
\(380\) 0 0
\(381\) −1.33591 −0.0684407
\(382\) 0 0
\(383\) −24.1407 −1.23353 −0.616767 0.787146i \(-0.711558\pi\)
−0.616767 + 0.787146i \(0.711558\pi\)
\(384\) 0 0
\(385\) 0.729571 0.0371824
\(386\) 0 0
\(387\) −2.03081 −0.103232
\(388\) 0 0
\(389\) 16.2953 0.826206 0.413103 0.910684i \(-0.364445\pi\)
0.413103 + 0.910684i \(0.364445\pi\)
\(390\) 0 0
\(391\) −8.69231 −0.439589
\(392\) 0 0
\(393\) 4.44091 0.224014
\(394\) 0 0
\(395\) 13.7541 0.692043
\(396\) 0 0
\(397\) 36.8139 1.84764 0.923819 0.382829i \(-0.125050\pi\)
0.923819 + 0.382829i \(0.125050\pi\)
\(398\) 0 0
\(399\) −0.274782 −0.0137563
\(400\) 0 0
\(401\) −23.6899 −1.18302 −0.591508 0.806299i \(-0.701467\pi\)
−0.591508 + 0.806299i \(0.701467\pi\)
\(402\) 0 0
\(403\) 8.73941 0.435341
\(404\) 0 0
\(405\) −1.26951 −0.0630827
\(406\) 0 0
\(407\) 11.5622 0.573118
\(408\) 0 0
\(409\) −0.102363 −0.00506154 −0.00253077 0.999997i \(-0.500806\pi\)
−0.00253077 + 0.999997i \(0.500806\pi\)
\(410\) 0 0
\(411\) −1.64681 −0.0812311
\(412\) 0 0
\(413\) −2.20891 −0.108693
\(414\) 0 0
\(415\) −14.4197 −0.707836
\(416\) 0 0
\(417\) −4.12568 −0.202035
\(418\) 0 0
\(419\) −25.7326 −1.25712 −0.628560 0.777761i \(-0.716355\pi\)
−0.628560 + 0.777761i \(0.716355\pi\)
\(420\) 0 0
\(421\) −18.0014 −0.877334 −0.438667 0.898650i \(-0.644549\pi\)
−0.438667 + 0.898650i \(0.644549\pi\)
\(422\) 0 0
\(423\) 17.1842 0.835524
\(424\) 0 0
\(425\) 17.1867 0.833675
\(426\) 0 0
\(427\) 1.11180 0.0538036
\(428\) 0 0
\(429\) −4.30350 −0.207775
\(430\) 0 0
\(431\) −27.6392 −1.33133 −0.665667 0.746249i \(-0.731853\pi\)
−0.665667 + 0.746249i \(0.731853\pi\)
\(432\) 0 0
\(433\) 38.5219 1.85124 0.925621 0.378451i \(-0.123543\pi\)
0.925621 + 0.378451i \(0.123543\pi\)
\(434\) 0 0
\(435\) −10.9539 −0.525198
\(436\) 0 0
\(437\) −1.94338 −0.0929643
\(438\) 0 0
\(439\) 23.1302 1.10394 0.551971 0.833863i \(-0.313876\pi\)
0.551971 + 0.833863i \(0.313876\pi\)
\(440\) 0 0
\(441\) 11.6181 0.553241
\(442\) 0 0
\(443\) −19.2436 −0.914290 −0.457145 0.889392i \(-0.651128\pi\)
−0.457145 + 0.889392i \(0.651128\pi\)
\(444\) 0 0
\(445\) 9.90505 0.469544
\(446\) 0 0
\(447\) 18.7473 0.886716
\(448\) 0 0
\(449\) 11.0578 0.521851 0.260926 0.965359i \(-0.415972\pi\)
0.260926 + 0.965359i \(0.415972\pi\)
\(450\) 0 0
\(451\) −7.90812 −0.372379
\(452\) 0 0
\(453\) −16.4988 −0.775183
\(454\) 0 0
\(455\) −0.337372 −0.0158162
\(456\) 0 0
\(457\) −4.08310 −0.190999 −0.0954996 0.995429i \(-0.530445\pi\)
−0.0954996 + 0.995429i \(0.530445\pi\)
\(458\) 0 0
\(459\) 24.0761 1.12378
\(460\) 0 0
\(461\) 7.59017 0.353509 0.176755 0.984255i \(-0.443440\pi\)
0.176755 + 0.984255i \(0.443440\pi\)
\(462\) 0 0
\(463\) −37.2637 −1.73179 −0.865894 0.500227i \(-0.833250\pi\)
−0.865894 + 0.500227i \(0.833250\pi\)
\(464\) 0 0
\(465\) −8.23920 −0.382084
\(466\) 0 0
\(467\) 8.92480 0.412991 0.206495 0.978448i \(-0.433794\pi\)
0.206495 + 0.978448i \(0.433794\pi\)
\(468\) 0 0
\(469\) −1.03024 −0.0475720
\(470\) 0 0
\(471\) 4.30196 0.198224
\(472\) 0 0
\(473\) 3.44974 0.158619
\(474\) 0 0
\(475\) 3.84249 0.176306
\(476\) 0 0
\(477\) −13.9929 −0.640690
\(478\) 0 0
\(479\) 19.6241 0.896650 0.448325 0.893871i \(-0.352021\pi\)
0.448325 + 0.893871i \(0.352021\pi\)
\(480\) 0 0
\(481\) −5.34666 −0.243787
\(482\) 0 0
\(483\) 0.534005 0.0242981
\(484\) 0 0
\(485\) −0.266277 −0.0120910
\(486\) 0 0
\(487\) 13.5448 0.613775 0.306887 0.951746i \(-0.400712\pi\)
0.306887 + 0.951746i \(0.400712\pi\)
\(488\) 0 0
\(489\) 4.59413 0.207754
\(490\) 0 0
\(491\) 9.58255 0.432454 0.216227 0.976343i \(-0.430625\pi\)
0.216227 + 0.976343i \(0.430625\pi\)
\(492\) 0 0
\(493\) 39.5369 1.78065
\(494\) 0 0
\(495\) −5.11732 −0.230006
\(496\) 0 0
\(497\) 0.663374 0.0297564
\(498\) 0 0
\(499\) −7.24834 −0.324480 −0.162240 0.986751i \(-0.551872\pi\)
−0.162240 + 0.986751i \(0.551872\pi\)
\(500\) 0 0
\(501\) 9.41213 0.420503
\(502\) 0 0
\(503\) −2.19200 −0.0977367 −0.0488683 0.998805i \(-0.515561\pi\)
−0.0488683 + 0.998805i \(0.515561\pi\)
\(504\) 0 0
\(505\) 18.0775 0.804440
\(506\) 0 0
\(507\) −12.9835 −0.576618
\(508\) 0 0
\(509\) 13.5765 0.601766 0.300883 0.953661i \(-0.402719\pi\)
0.300883 + 0.953661i \(0.402719\pi\)
\(510\) 0 0
\(511\) −0.0320774 −0.00141902
\(512\) 0 0
\(513\) 5.38280 0.237656
\(514\) 0 0
\(515\) 5.82002 0.256461
\(516\) 0 0
\(517\) −29.1909 −1.28381
\(518\) 0 0
\(519\) 0.404696 0.0177642
\(520\) 0 0
\(521\) −6.57594 −0.288097 −0.144049 0.989571i \(-0.546012\pi\)
−0.144049 + 0.989571i \(0.546012\pi\)
\(522\) 0 0
\(523\) 22.2161 0.971443 0.485722 0.874114i \(-0.338557\pi\)
0.485722 + 0.874114i \(0.338557\pi\)
\(524\) 0 0
\(525\) −1.05585 −0.0460810
\(526\) 0 0
\(527\) 29.7386 1.29543
\(528\) 0 0
\(529\) −19.2233 −0.835795
\(530\) 0 0
\(531\) 15.4936 0.672366
\(532\) 0 0
\(533\) 3.65691 0.158398
\(534\) 0 0
\(535\) −6.87605 −0.297278
\(536\) 0 0
\(537\) −17.5928 −0.759185
\(538\) 0 0
\(539\) −19.7357 −0.850076
\(540\) 0 0
\(541\) 22.5004 0.967368 0.483684 0.875243i \(-0.339298\pi\)
0.483684 + 0.875243i \(0.339298\pi\)
\(542\) 0 0
\(543\) 10.3045 0.442207
\(544\) 0 0
\(545\) −16.0068 −0.685656
\(546\) 0 0
\(547\) 24.1551 1.03280 0.516399 0.856348i \(-0.327272\pi\)
0.516399 + 0.856348i \(0.327272\pi\)
\(548\) 0 0
\(549\) −7.79830 −0.332823
\(550\) 0 0
\(551\) 8.83944 0.376573
\(552\) 0 0
\(553\) 3.04984 0.129692
\(554\) 0 0
\(555\) 5.04063 0.213963
\(556\) 0 0
\(557\) −21.4469 −0.908734 −0.454367 0.890815i \(-0.650135\pi\)
−0.454367 + 0.890815i \(0.650135\pi\)
\(558\) 0 0
\(559\) −1.59525 −0.0674717
\(560\) 0 0
\(561\) −14.6440 −0.618270
\(562\) 0 0
\(563\) −15.3594 −0.647322 −0.323661 0.946173i \(-0.604914\pi\)
−0.323661 + 0.946173i \(0.604914\pi\)
\(564\) 0 0
\(565\) 9.24256 0.388837
\(566\) 0 0
\(567\) −0.281503 −0.0118220
\(568\) 0 0
\(569\) 2.05613 0.0861973 0.0430986 0.999071i \(-0.486277\pi\)
0.0430986 + 0.999071i \(0.486277\pi\)
\(570\) 0 0
\(571\) −35.4718 −1.48445 −0.742225 0.670151i \(-0.766229\pi\)
−0.742225 + 0.670151i \(0.766229\pi\)
\(572\) 0 0
\(573\) 21.1871 0.885103
\(574\) 0 0
\(575\) −7.46741 −0.311413
\(576\) 0 0
\(577\) 14.6910 0.611595 0.305798 0.952097i \(-0.401077\pi\)
0.305798 + 0.952097i \(0.401077\pi\)
\(578\) 0 0
\(579\) −5.87311 −0.244078
\(580\) 0 0
\(581\) −3.19743 −0.132652
\(582\) 0 0
\(583\) 23.7698 0.984445
\(584\) 0 0
\(585\) 2.36638 0.0978376
\(586\) 0 0
\(587\) −1.71339 −0.0707193 −0.0353596 0.999375i \(-0.511258\pi\)
−0.0353596 + 0.999375i \(0.511258\pi\)
\(588\) 0 0
\(589\) 6.64877 0.273958
\(590\) 0 0
\(591\) −23.3722 −0.961402
\(592\) 0 0
\(593\) 20.3947 0.837509 0.418754 0.908100i \(-0.362467\pi\)
0.418754 + 0.908100i \(0.362467\pi\)
\(594\) 0 0
\(595\) −1.14801 −0.0470639
\(596\) 0 0
\(597\) 0.358212 0.0146607
\(598\) 0 0
\(599\) −21.9447 −0.896637 −0.448318 0.893874i \(-0.647977\pi\)
−0.448318 + 0.893874i \(0.647977\pi\)
\(600\) 0 0
\(601\) −3.41894 −0.139461 −0.0697306 0.997566i \(-0.522214\pi\)
−0.0697306 + 0.997566i \(0.522214\pi\)
\(602\) 0 0
\(603\) 7.22625 0.294276
\(604\) 0 0
\(605\) −3.14180 −0.127732
\(606\) 0 0
\(607\) 23.1235 0.938555 0.469278 0.883051i \(-0.344514\pi\)
0.469278 + 0.883051i \(0.344514\pi\)
\(608\) 0 0
\(609\) −2.42892 −0.0984247
\(610\) 0 0
\(611\) 13.4986 0.546095
\(612\) 0 0
\(613\) −46.3763 −1.87312 −0.936561 0.350504i \(-0.886010\pi\)
−0.936561 + 0.350504i \(0.886010\pi\)
\(614\) 0 0
\(615\) −3.44760 −0.139021
\(616\) 0 0
\(617\) −25.2123 −1.01501 −0.507504 0.861649i \(-0.669432\pi\)
−0.507504 + 0.861649i \(0.669432\pi\)
\(618\) 0 0
\(619\) 37.2031 1.49532 0.747660 0.664082i \(-0.231178\pi\)
0.747660 + 0.664082i \(0.231178\pi\)
\(620\) 0 0
\(621\) −10.4608 −0.419778
\(622\) 0 0
\(623\) 2.19635 0.0879948
\(624\) 0 0
\(625\) 8.97722 0.359089
\(626\) 0 0
\(627\) −3.27402 −0.130752
\(628\) 0 0
\(629\) −18.1937 −0.725429
\(630\) 0 0
\(631\) −40.8412 −1.62586 −0.812931 0.582360i \(-0.802129\pi\)
−0.812931 + 0.582360i \(0.802129\pi\)
\(632\) 0 0
\(633\) −16.5036 −0.655959
\(634\) 0 0
\(635\) −1.24783 −0.0495188
\(636\) 0 0
\(637\) 9.12627 0.361596
\(638\) 0 0
\(639\) −4.65300 −0.184070
\(640\) 0 0
\(641\) 5.87446 0.232027 0.116014 0.993248i \(-0.462988\pi\)
0.116014 + 0.993248i \(0.462988\pi\)
\(642\) 0 0
\(643\) 21.3344 0.841347 0.420673 0.907212i \(-0.361794\pi\)
0.420673 + 0.907212i \(0.361794\pi\)
\(644\) 0 0
\(645\) 1.50394 0.0592176
\(646\) 0 0
\(647\) −48.1625 −1.89346 −0.946732 0.322024i \(-0.895637\pi\)
−0.946732 + 0.322024i \(0.895637\pi\)
\(648\) 0 0
\(649\) −26.3191 −1.03311
\(650\) 0 0
\(651\) −1.82696 −0.0716043
\(652\) 0 0
\(653\) 11.3441 0.443929 0.221965 0.975055i \(-0.428753\pi\)
0.221965 + 0.975055i \(0.428753\pi\)
\(654\) 0 0
\(655\) 4.14813 0.162081
\(656\) 0 0
\(657\) 0.224995 0.00877791
\(658\) 0 0
\(659\) −24.6320 −0.959526 −0.479763 0.877398i \(-0.659277\pi\)
−0.479763 + 0.877398i \(0.659277\pi\)
\(660\) 0 0
\(661\) −47.3409 −1.84135 −0.920673 0.390335i \(-0.872359\pi\)
−0.920673 + 0.390335i \(0.872359\pi\)
\(662\) 0 0
\(663\) 6.77175 0.262993
\(664\) 0 0
\(665\) −0.256666 −0.00995308
\(666\) 0 0
\(667\) −17.1784 −0.665149
\(668\) 0 0
\(669\) −14.8148 −0.572772
\(670\) 0 0
\(671\) 13.2470 0.511396
\(672\) 0 0
\(673\) −35.1767 −1.35596 −0.677981 0.735079i \(-0.737145\pi\)
−0.677981 + 0.735079i \(0.737145\pi\)
\(674\) 0 0
\(675\) 20.6834 0.796103
\(676\) 0 0
\(677\) 2.71085 0.104186 0.0520932 0.998642i \(-0.483411\pi\)
0.0520932 + 0.998642i \(0.483411\pi\)
\(678\) 0 0
\(679\) −0.0590444 −0.00226592
\(680\) 0 0
\(681\) −21.2499 −0.814300
\(682\) 0 0
\(683\) 48.4820 1.85511 0.927557 0.373682i \(-0.121905\pi\)
0.927557 + 0.373682i \(0.121905\pi\)
\(684\) 0 0
\(685\) −1.53824 −0.0587730
\(686\) 0 0
\(687\) −5.45128 −0.207979
\(688\) 0 0
\(689\) −10.9917 −0.418752
\(690\) 0 0
\(691\) −41.3295 −1.57225 −0.786124 0.618068i \(-0.787915\pi\)
−0.786124 + 0.618068i \(0.787915\pi\)
\(692\) 0 0
\(693\) −1.13472 −0.0431043
\(694\) 0 0
\(695\) −3.85368 −0.146178
\(696\) 0 0
\(697\) 12.4438 0.471342
\(698\) 0 0
\(699\) −4.92755 −0.186377
\(700\) 0 0
\(701\) 15.9692 0.603147 0.301573 0.953443i \(-0.402488\pi\)
0.301573 + 0.953443i \(0.402488\pi\)
\(702\) 0 0
\(703\) −4.06763 −0.153414
\(704\) 0 0
\(705\) −12.7260 −0.479288
\(706\) 0 0
\(707\) 4.00852 0.150756
\(708\) 0 0
\(709\) −1.27532 −0.0478957 −0.0239478 0.999713i \(-0.507624\pi\)
−0.0239478 + 0.999713i \(0.507624\pi\)
\(710\) 0 0
\(711\) −21.3920 −0.802263
\(712\) 0 0
\(713\) −12.9211 −0.483898
\(714\) 0 0
\(715\) −4.01978 −0.150331
\(716\) 0 0
\(717\) 23.4860 0.877100
\(718\) 0 0
\(719\) −38.6245 −1.44045 −0.720225 0.693740i \(-0.755962\pi\)
−0.720225 + 0.693740i \(0.755962\pi\)
\(720\) 0 0
\(721\) 1.29053 0.0480619
\(722\) 0 0
\(723\) 29.4722 1.09608
\(724\) 0 0
\(725\) 33.9655 1.26145
\(726\) 0 0
\(727\) −3.96945 −0.147219 −0.0736095 0.997287i \(-0.523452\pi\)
−0.0736095 + 0.997287i \(0.523452\pi\)
\(728\) 0 0
\(729\) 20.5744 0.762015
\(730\) 0 0
\(731\) −5.42832 −0.200774
\(732\) 0 0
\(733\) 23.9821 0.885800 0.442900 0.896571i \(-0.353950\pi\)
0.442900 + 0.896571i \(0.353950\pi\)
\(734\) 0 0
\(735\) −8.60391 −0.317360
\(736\) 0 0
\(737\) −12.2753 −0.452165
\(738\) 0 0
\(739\) 34.2553 1.26010 0.630051 0.776554i \(-0.283034\pi\)
0.630051 + 0.776554i \(0.283034\pi\)
\(740\) 0 0
\(741\) 1.51399 0.0556177
\(742\) 0 0
\(743\) 31.0058 1.13749 0.568746 0.822513i \(-0.307429\pi\)
0.568746 + 0.822513i \(0.307429\pi\)
\(744\) 0 0
\(745\) 17.5113 0.641564
\(746\) 0 0
\(747\) 22.4273 0.820571
\(748\) 0 0
\(749\) −1.52470 −0.0557112
\(750\) 0 0
\(751\) 36.0417 1.31518 0.657590 0.753376i \(-0.271576\pi\)
0.657590 + 0.753376i \(0.271576\pi\)
\(752\) 0 0
\(753\) −33.3849 −1.21661
\(754\) 0 0
\(755\) −15.4111 −0.560867
\(756\) 0 0
\(757\) 26.6253 0.967712 0.483856 0.875148i \(-0.339236\pi\)
0.483856 + 0.875148i \(0.339236\pi\)
\(758\) 0 0
\(759\) 6.36265 0.230950
\(760\) 0 0
\(761\) −40.0064 −1.45023 −0.725115 0.688628i \(-0.758213\pi\)
−0.725115 + 0.688628i \(0.758213\pi\)
\(762\) 0 0
\(763\) −3.54935 −0.128495
\(764\) 0 0
\(765\) 8.05233 0.291133
\(766\) 0 0
\(767\) 12.1706 0.439455
\(768\) 0 0
\(769\) 19.1147 0.689295 0.344648 0.938732i \(-0.387998\pi\)
0.344648 + 0.938732i \(0.387998\pi\)
\(770\) 0 0
\(771\) 5.87658 0.211640
\(772\) 0 0
\(773\) 42.2244 1.51871 0.759353 0.650679i \(-0.225516\pi\)
0.759353 + 0.650679i \(0.225516\pi\)
\(774\) 0 0
\(775\) 25.5479 0.917706
\(776\) 0 0
\(777\) 1.11771 0.0400977
\(778\) 0 0
\(779\) 2.78211 0.0996793
\(780\) 0 0
\(781\) 7.90408 0.282830
\(782\) 0 0
\(783\) 47.5809 1.70040
\(784\) 0 0
\(785\) 4.01833 0.143421
\(786\) 0 0
\(787\) 43.6009 1.55420 0.777102 0.629375i \(-0.216689\pi\)
0.777102 + 0.629375i \(0.216689\pi\)
\(788\) 0 0
\(789\) 13.4707 0.479571
\(790\) 0 0
\(791\) 2.04945 0.0728699
\(792\) 0 0
\(793\) −6.12575 −0.217532
\(794\) 0 0
\(795\) 10.3626 0.367524
\(796\) 0 0
\(797\) −14.6901 −0.520351 −0.260176 0.965561i \(-0.583780\pi\)
−0.260176 + 0.965561i \(0.583780\pi\)
\(798\) 0 0
\(799\) 45.9332 1.62500
\(800\) 0 0
\(801\) −15.4055 −0.544327
\(802\) 0 0
\(803\) −0.382201 −0.0134876
\(804\) 0 0
\(805\) 0.498799 0.0175803
\(806\) 0 0
\(807\) 0.188244 0.00662649
\(808\) 0 0
\(809\) −4.76004 −0.167354 −0.0836771 0.996493i \(-0.526666\pi\)
−0.0836771 + 0.996493i \(0.526666\pi\)
\(810\) 0 0
\(811\) 14.6172 0.513279 0.256640 0.966507i \(-0.417385\pi\)
0.256640 + 0.966507i \(0.417385\pi\)
\(812\) 0 0
\(813\) 1.16582 0.0408872
\(814\) 0 0
\(815\) 4.29125 0.150316
\(816\) 0 0
\(817\) −1.21363 −0.0424596
\(818\) 0 0
\(819\) 0.524721 0.0183352
\(820\) 0 0
\(821\) −32.3079 −1.12755 −0.563776 0.825928i \(-0.690652\pi\)
−0.563776 + 0.825928i \(0.690652\pi\)
\(822\) 0 0
\(823\) −9.79982 −0.341600 −0.170800 0.985306i \(-0.554635\pi\)
−0.170800 + 0.985306i \(0.554635\pi\)
\(824\) 0 0
\(825\) −12.5804 −0.437993
\(826\) 0 0
\(827\) 11.2066 0.389691 0.194845 0.980834i \(-0.437579\pi\)
0.194845 + 0.980834i \(0.437579\pi\)
\(828\) 0 0
\(829\) −45.4323 −1.57793 −0.788965 0.614438i \(-0.789383\pi\)
−0.788965 + 0.614438i \(0.789383\pi\)
\(830\) 0 0
\(831\) −3.62619 −0.125791
\(832\) 0 0
\(833\) 31.0550 1.07599
\(834\) 0 0
\(835\) 8.79160 0.304246
\(836\) 0 0
\(837\) 35.7890 1.23705
\(838\) 0 0
\(839\) −16.8997 −0.583441 −0.291721 0.956504i \(-0.594228\pi\)
−0.291721 + 0.956504i \(0.594228\pi\)
\(840\) 0 0
\(841\) 49.1356 1.69433
\(842\) 0 0
\(843\) 28.4191 0.978807
\(844\) 0 0
\(845\) −12.1275 −0.417200
\(846\) 0 0
\(847\) −0.696664 −0.0239377
\(848\) 0 0
\(849\) 15.9059 0.545890
\(850\) 0 0
\(851\) 7.90494 0.270978
\(852\) 0 0
\(853\) −37.7943 −1.29405 −0.647027 0.762467i \(-0.723988\pi\)
−0.647027 + 0.762467i \(0.723988\pi\)
\(854\) 0 0
\(855\) 1.80029 0.0615687
\(856\) 0 0
\(857\) −38.8619 −1.32750 −0.663749 0.747956i \(-0.731036\pi\)
−0.663749 + 0.747956i \(0.731036\pi\)
\(858\) 0 0
\(859\) 41.1795 1.40503 0.702513 0.711671i \(-0.252061\pi\)
0.702513 + 0.711671i \(0.252061\pi\)
\(860\) 0 0
\(861\) −0.764473 −0.0260532
\(862\) 0 0
\(863\) −25.5909 −0.871125 −0.435562 0.900159i \(-0.643450\pi\)
−0.435562 + 0.900159i \(0.643450\pi\)
\(864\) 0 0
\(865\) 0.378015 0.0128529
\(866\) 0 0
\(867\) 3.46216 0.117581
\(868\) 0 0
\(869\) 36.3387 1.23271
\(870\) 0 0
\(871\) 5.67639 0.192337
\(872\) 0 0
\(873\) 0.414146 0.0140167
\(874\) 0 0
\(875\) −2.26957 −0.0767253
\(876\) 0 0
\(877\) 14.4790 0.488920 0.244460 0.969659i \(-0.421389\pi\)
0.244460 + 0.969659i \(0.421389\pi\)
\(878\) 0 0
\(879\) −9.49879 −0.320386
\(880\) 0 0
\(881\) 48.3407 1.62864 0.814320 0.580416i \(-0.197110\pi\)
0.814320 + 0.580416i \(0.197110\pi\)
\(882\) 0 0
\(883\) 17.2669 0.581077 0.290539 0.956863i \(-0.406166\pi\)
0.290539 + 0.956863i \(0.406166\pi\)
\(884\) 0 0
\(885\) −11.4740 −0.385694
\(886\) 0 0
\(887\) 51.0650 1.71460 0.857298 0.514820i \(-0.172141\pi\)
0.857298 + 0.514820i \(0.172141\pi\)
\(888\) 0 0
\(889\) −0.276695 −0.00928006
\(890\) 0 0
\(891\) −3.35410 −0.112366
\(892\) 0 0
\(893\) 10.2695 0.343655
\(894\) 0 0
\(895\) −16.4329 −0.549292
\(896\) 0 0
\(897\) −2.94225 −0.0982388
\(898\) 0 0
\(899\) 58.7714 1.96014
\(900\) 0 0
\(901\) −37.4028 −1.24607
\(902\) 0 0
\(903\) 0.333484 0.0110977
\(904\) 0 0
\(905\) 9.62510 0.319949
\(906\) 0 0
\(907\) 30.0974 0.999367 0.499684 0.866208i \(-0.333450\pi\)
0.499684 + 0.866208i \(0.333450\pi\)
\(908\) 0 0
\(909\) −28.1164 −0.932561
\(910\) 0 0
\(911\) −9.15413 −0.303290 −0.151645 0.988435i \(-0.548457\pi\)
−0.151645 + 0.988435i \(0.548457\pi\)
\(912\) 0 0
\(913\) −38.0973 −1.26084
\(914\) 0 0
\(915\) 5.77514 0.190920
\(916\) 0 0
\(917\) 0.919807 0.0303747
\(918\) 0 0
\(919\) −7.49270 −0.247161 −0.123581 0.992335i \(-0.539438\pi\)
−0.123581 + 0.992335i \(0.539438\pi\)
\(920\) 0 0
\(921\) −20.1037 −0.662440
\(922\) 0 0
\(923\) −3.65505 −0.120307
\(924\) 0 0
\(925\) −15.6299 −0.513906
\(926\) 0 0
\(927\) −9.05198 −0.297306
\(928\) 0 0
\(929\) −12.0792 −0.396306 −0.198153 0.980171i \(-0.563494\pi\)
−0.198153 + 0.980171i \(0.563494\pi\)
\(930\) 0 0
\(931\) 6.94309 0.227550
\(932\) 0 0
\(933\) 12.1235 0.396905
\(934\) 0 0
\(935\) −13.6785 −0.447336
\(936\) 0 0
\(937\) 16.6732 0.544690 0.272345 0.962200i \(-0.412201\pi\)
0.272345 + 0.962200i \(0.412201\pi\)
\(938\) 0 0
\(939\) 16.5283 0.539380
\(940\) 0 0
\(941\) −29.5263 −0.962529 −0.481265 0.876575i \(-0.659822\pi\)
−0.481265 + 0.876575i \(0.659822\pi\)
\(942\) 0 0
\(943\) −5.40668 −0.176066
\(944\) 0 0
\(945\) −1.38158 −0.0449428
\(946\) 0 0
\(947\) −24.2099 −0.786717 −0.393358 0.919385i \(-0.628687\pi\)
−0.393358 + 0.919385i \(0.628687\pi\)
\(948\) 0 0
\(949\) 0.176739 0.00573720
\(950\) 0 0
\(951\) −18.8287 −0.610561
\(952\) 0 0
\(953\) 18.5304 0.600258 0.300129 0.953899i \(-0.402970\pi\)
0.300129 + 0.953899i \(0.402970\pi\)
\(954\) 0 0
\(955\) 19.7903 0.640398
\(956\) 0 0
\(957\) −28.9405 −0.935513
\(958\) 0 0
\(959\) −0.341089 −0.0110143
\(960\) 0 0
\(961\) 13.2062 0.426005
\(962\) 0 0
\(963\) 10.6945 0.344624
\(964\) 0 0
\(965\) −5.48590 −0.176597
\(966\) 0 0
\(967\) 54.2821 1.74559 0.872797 0.488084i \(-0.162304\pi\)
0.872797 + 0.488084i \(0.162304\pi\)
\(968\) 0 0
\(969\) 5.15181 0.165500
\(970\) 0 0
\(971\) −7.69284 −0.246875 −0.123437 0.992352i \(-0.539392\pi\)
−0.123437 + 0.992352i \(0.539392\pi\)
\(972\) 0 0
\(973\) −0.854516 −0.0273945
\(974\) 0 0
\(975\) 5.81749 0.186309
\(976\) 0 0
\(977\) −32.8587 −1.05124 −0.525621 0.850719i \(-0.676167\pi\)
−0.525621 + 0.850719i \(0.676167\pi\)
\(978\) 0 0
\(979\) 26.1694 0.836378
\(980\) 0 0
\(981\) 24.8957 0.794858
\(982\) 0 0
\(983\) −10.4144 −0.332169 −0.166084 0.986112i \(-0.553112\pi\)
−0.166084 + 0.986112i \(0.553112\pi\)
\(984\) 0 0
\(985\) −21.8313 −0.695602
\(986\) 0 0
\(987\) −2.82186 −0.0898209
\(988\) 0 0
\(989\) 2.35855 0.0749974
\(990\) 0 0
\(991\) 62.1164 1.97319 0.986596 0.163183i \(-0.0521761\pi\)
0.986596 + 0.163183i \(0.0521761\pi\)
\(992\) 0 0
\(993\) 17.2908 0.548706
\(994\) 0 0
\(995\) 0.334596 0.0106074
\(996\) 0 0
\(997\) 5.60362 0.177468 0.0887341 0.996055i \(-0.471718\pi\)
0.0887341 + 0.996055i \(0.471718\pi\)
\(998\) 0 0
\(999\) −21.8952 −0.692735
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4864.2.a.bs.1.8 10
4.3 odd 2 4864.2.a.bt.1.4 10
8.3 odd 2 inner 4864.2.a.bs.1.7 10
8.5 even 2 4864.2.a.bt.1.3 10
16.3 odd 4 2432.2.c.j.1217.7 20
16.5 even 4 2432.2.c.j.1217.8 yes 20
16.11 odd 4 2432.2.c.j.1217.14 yes 20
16.13 even 4 2432.2.c.j.1217.13 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.j.1217.7 20 16.3 odd 4
2432.2.c.j.1217.8 yes 20 16.5 even 4
2432.2.c.j.1217.13 yes 20 16.13 even 4
2432.2.c.j.1217.14 yes 20 16.11 odd 4
4864.2.a.bs.1.7 10 8.3 odd 2 inner
4864.2.a.bs.1.8 10 1.1 even 1 trivial
4864.2.a.bt.1.3 10 8.5 even 2
4864.2.a.bt.1.4 10 4.3 odd 2